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# sstvars

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The goal of `sstvars` is to provide a comprehensive toolkit for
(penalized and non-penalized) maximum likelihood (ML) estimation and
analysis of reduced form and structural smooth transition vector
autoregressive (STVAR) models (including threshold VAR models). Various
transition weight functions, conditional distributions, and
identification methods are accommodated. Also constrained ML estimation
is supported with constraints on the autoregressive parameters,
regimewise means, weight parameters, and the impact matrix. See the
vignette for a more detailed description of the package.

## Installation

You can install the released version of gmvarkit from
[CRAN](https://CRAN.R-project.org/package=sstvars) with:

``` r
install.packages("sstvars")
```

You can install the development version of sstvars from
[GitHub](https://github.com/) with:

``` r
# install.packages("devtools")
devtools::install_github("saviviro/sstvars")
```

## Example

This is a basic example on how to use `sstvars` in time series analysis.
The estimation process is computationally demanding and takes advantage
of parallel computing. After estimating the model, it is shown by simple
examples how to conduct some further analysis.

``` r
# These examples use the data 'gdpdef' which comes with the package, and contains the quarterly percentage growth rate
# of real U.S. GDP and quarterly percentage growth rate of U.S. GDP implicit price deflator, covering the period 
# from 1959Q1 to 2019Q4.
data(gdpdef, package="sstvars")

# Some of the below examples are computationally demanding. Running them all will take approximately 10 minutes.

### Reduced form STVAR models ###

# Estimate a reduced form two-regime Student's t STVAR p=2 model with threshold transition weight function using the first
# lag of the first variable (GDP) as the switching variable. The below estimation is based on two estimation
# rounds with seeds set for reproducibility.
# (IMPORTANT: typically empirical applications require more estimation rounds, e.g., tens, hundreds or even thousand, depending
# on the size of the model, and with the two-phase procedure often much more).
fit <- fitSTVAR(gdpdef, p=2, M=2, weight_function="threshold", weightfun_pars=c(2, 1), cond_dist="Student",
                estim_method="three-phase", nrounds=2, ncores=2, seeds=1:2)
                
# Information on the estimated model:
plot(fit) # Plot the estimated transition weight function with data
summary(fit) # Summary printoout of the estimated model
get_foc(fit) # The first order condition (gradient of the log-likelihood function)
get_soc(fit) # The second order condition (eigenvalues of approximated Hessian)
profile_logliks(fit) # Plot profile log-likelihood functions about the estimate

# See also the functions alt_stvar and filter_estimates.

# Check the stationarity condition for the estimated model, i.e., that the 
# upper bound of the joint spectral radius is less than one:
bound_JSR(fit, epsilon=0.1, ncores=2) # Adjust epsilon for a tighter bound
# NOTE: For models that are not small, our implementation is not computationally
# efficient enough. The MATLAB Toolbox JSR (by R. Jungers) for large can be used
# larger models. 

# Estimate the above model but with the autoregressive matrices restricted to be equal in both regimes
# (so that only the intercepts and the conditional covariance matrix vary in time):
C_mat <- rbind(diag(2*2^2), diag(2*2^2))
fitc <- fitSTVAR(gdpdef, p=2, M=2, weight_function="threshold", weightfun_pars=c(2, 1), cond_dist="Student",
                 AR_constraints=C_mat, nrounds=2, ncores=2, seeds=1:2)

# Estimate the above model but with the autoregressive matrices and unconditional means restricted to be equal
# in both regimes (so that only the conditional covariance matrix varies in time), two-phase estimation 
# is used because mean-constraints are not supported in the three-phase estimation:
fitcm <- fitSTVAR(gdpdef, p=2, M=2, weight_function="threshold", weightfun_pars=c(2, 1), cond_dist="Student",
                  AR_constraints=C_mat, mean_constraints=list(1:2), nrounds=2, ncores=2, seeds=1:2)

# Estimate the above logistic STVAR model without constraints on the autoregressive parameters but with the 
# the location parameter constrained to 1 and scale parameter unconstrained. Two-phase estimation is used because
# this type of weight constraints  are not supported in the three-phase estimation:
fitw <- fitSTVAR(gdpdef, p=2, M=2, weight_function="logistic", weightfun_pars=c(2, 1), cond_dist="Student",
                 weight_constraints=list(R=matrix(c(0, 1), nrow=2), r=c(1, 0)), nrounds=2, ncores=2, seeds=1:2)

# The constraints can be tested with the functions LR_test, Wald_test, and Rao_test.

# Residual based model diagnostics:
diagnostic_plot(fit, type="series", resid_type="standardized") # Standardized residual time series
diagnostic_plot(fit, type="ac", resid_type="raw") # Autocorrelation function of unstandardized residuals
diagnostic_plot(fit, type="ch", resid_type="standardized") # Autocorrelation function of squared standardized residuals
diagnostic_plot(fit, type="dist", resid_type="standardized") # Histograms and Q-Q plots of standardized residuals

Portmanteau_test(fit, nlags=20, which_test="autocorr") # Portmanteau test for remaining autocorrelation
Portmanteau_test(fit, nlags=20, which_test="het.sked") # Portmanteau test applied for testing cond. het.kedasticity

# Simulate a sample path from the estimated model, initial drawn from the first regime:
sim <- simulate(fit, nsim=100, init_regime=1)

# Forecast future values of the process:
pred <- predict(fit, nsteps=10, ci=c(0.95, 0.80))
plot(pred)


### Structural STVAR models ###

# stvars implements two identification methods: recursive identification and
# identification by heteroskedasticity. The structural models are estimated 
# based on preliminary estimates from a reduced form model. If the structural model
# is not overidentifying, the model is merely reparametrized and no estimation is
# required (recursively identified models are just reduced form models marked as structural). 

# Identify the above threshold VAR model by recursive identification:
fitrec <- fitSSTVAR(fit, identification="recursive")
fitrec

# Identify the above threshold VAR model by heteroskedasticity:
fithet <- fitSSTVAR(fit, identification="heteroskedasticity")
fithet

# Identification by non-Gaussianity available for models with independent Student's t distribution
# or independent skewed t distribution as the conditional distribution. The reduced form model is
# then readily identified by non-Gaussianity. Estimate a reduced form model identified by
# non-Gaussianity with independent Student's t shocks: 
fitindt <- fitSTVAR(gdpdef, p=1, M=2, weight_function="logistic", weightfun_pars=c(1, 1),
                    cond_dist="ind_Student", nrounds=2, ncores=2, seeds=1:2)
fitindt

# Impose overidentying constraint with the argument B_constraints by estimating
# with fitSSTVARs:
fitindtb <- fitSSTVAR(fitindt, identification="non-Gaussianity",
                      B_constraints=matrix(c(NA, NA, 0, NA), nrow=2))

# Reorder the columns of the impact matrix of fithet to the reverse ordering:
fithet <- reorder_B_columns(fithet, perm=c(2, 1))
fithet

# Change all signs of the first column of the impact matrix of fithet:
fithet <- swap_B_signs(fithet, which_to_swap=1)
fithet

# Structural models based on different orderings of signs of the columns of any
# single B_1,...B_M can be estimated by specifying the arguments B_pm_reg, B_perm,
# and B_signs in fitSSTVAR. 


# Estimate the generalized impulse response function (GIRF) for the recursively
# identified model to one-standard-error positive shocks with the starting values
# generated form the first regime, N=30 steps ahead and 95% confidence intervals 
# that reflect uncertainty about the initial value within the regime:
girf1 <- GIRF(fitrec, which_shocks=1:2, shock_size=1, N=30, init_regime=1, ci=0.95)
plot(girf1)

# Estimate the above GIRF but instead of drawing initial values form the first regime,
# use tha last p observations of the data as the initial values:
girf2 <- GIRF(fitrec, which_shocks=1:2, shock_size=1, N=30, init_values=fitrec$data)
plot(girf2)

# Estimate the generalized impulse response function (GIRF) for the recursively
# identified model to two-standard-error negative shocks with the starting values
# generated form the second regime, N=30 steps ahead and 95% confidence intervals 
# that reflect uncertainty about the initial value within the regime. Also, scale
# the responses to the first shock to correspond to a 0.3 increase of the first variable.
# Moreover, accumulate the responses of the second variable.
girf3 <- GIRF(fitrec, which_shocks=1:2, shock_size=-2, N=30, init_regime=2, 
              scale=c(1, 1, 0.3), which_cumulative=2, ci=0.95)
plot(girf3)

# Estimate the generalized forecast error variance decomposition (GFEVD) for the 
# recursively identified model with the initial values being all possible length p
# histories in the data, N=30 steps ahead to one-standard-error positive shocks. 
gfevd1 <- GFEVD(fitrec, shock_size=1, N=30, initval_type="data")
plot(gfevd1)

# Estimate the GFEVD for the recursively identified model with the initial values
# being all possible length p histories in the data AND the signs and sizes of the
# corresponding shocks being the identified structural shocks recovered from the
# fitted model.
gfevd2 <- GFEVD(fitrec, N=30, use_data_shocks=TRUE)
plot(gfevd2)

# Plot time series of the impact response GFEVDs to the first shock to examine 
# the contribution of the first shocks to the forecast error variances at impact
# at each point of time:
plot(gfevd2, data_shock_pars=c(1, 0))

# Estimate the linear impulse response function for the recursively identified
# STVAR model based on the first regime, the responses of the second variable
# accumulated:
irf1 <- linear_IRF(fitrec, N=30, regime=1, which_cumulative=2)
plot(irf1)
# The above model is nonlinear, so confidence bounds (that reflect the uncertainty
# about the parameter estimate) are not available.

# Bootstrapped confidence bounds can be calculated for models with time-invariant
# autoregressive coeffients, e.g., the restricted model fitcm estimated above. 
# Identify the shocks if fitcm by heteroskedasticity:
fitcmhet <- fitSSTVAR(fitcm, identification="heteroskedasticity")
fitcmhet

# Estimate the linear impulse reponse function for fitcmhet with bootstrapped
# 95% confidence bounds that reflect uncertainty about the parameter estimates:
irf2 <- linear_IRF(fitcmhet, N=30, ci=0.95, bootstrap_reps=250)
plot(irf2)

# Compute historical decompositions of the variables in the model fitrec (see Wong 2018):
histdec <- hist_decomp(fitrec)
plot(histdec)

# Simulate historical counterfactuals with the function cfact_hist,
# counterfactual forecast scenarios with the function cfact_fore, and
# counterfactual GIRFs with the function cfact_girf.
```

## References

- Anderson H., Vahid F. 1998. Testing multiple equation systems for
  common nonlinear components. *Journal of Econometrics*, **84**:1,
  1-36.
- Hubrich K., Teräsvirta. T. 2013. Thresholds and Smooth Transitions in
  Vector Autoregressive Models. *CREATES Research Paper 2013-18, Aarhus
  University.*
- Kheifets I., Saikkonen P. 2020. Stationarity and ergodicity of vector
  STAR models. *Econometrics Review*, **39**:407-414, 1311-1324.
- Koop G., Pesaran M.H., Potter S.M. 1996. Impulse response analysis in
  nonlinear multivariate models. *Journal of Econometrics*, **74**:1,
  119-147.
- Lanne M., Virolainen S. 2025. A Gaussian smooth transition vector
  autoregressive model: An application to the macroeconomic effects of
  severe weather shocks. *Journal of Economic Dynamics and Control*,
  **178**, 105162.
- Virolainen S. in press. Identification by non-Gaussianity in
  structural smooth transition vector autoregressive models.
  *Econometric Reviews*.
- Wong B. 2018. Historical decomposition for nonlinear vector
  autoregressive models. *CAMA Working Paper No. 62/2017, available as
  SSRN:3057759*.
